Spin-Gap Phases in Tomonaga-Luttinger Liquids Masaaki Nakamura∗ Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo 106-8666, Japan
arXiv:cond-mat/9904381v1 [cond-mat.str-el] 27 Apr 1999
Atsuhiro Kitazawa† and Kiyohide Nomura‡ Department of Physics, Kyushu University, Fukuoka 812-8581, Japan (December 14, 1998)
We give the details of the analysis for critical properties of spin-gap phases in one-dimensional lattice electron models. In the Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and the triplet excitation spectra, using the c = 1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the t-J model, and the t-J-J ′ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits. 71.10.Hf,71.30.+h,74.20.Mn
Since the gap opens slowly near the critical point, it is very difficult to find the critical point using conventional finite-size scaling method. In this paper, we give a remedy for this problem3,4 . The many-body problem is often simplified by using the notion of universality. Generally, 1D electron systems belong to the universality class of Tomonaga-Luttinger (TL) liquids5,6,7,8,9 which are characterized by gapless charge and spin excitations and power-law decay of correlation functions. This behavior can be described by the bosonization theory or the c = 1 conformal field theory (CFT). In this scheme, the phase transition to the spin-gap phase is understood as an instability caused by the backward scattering process using the renormalization group technique10 , and a spin gap opens when the backward scattering turns from repulsive to attractive. This transition point is equivalent to the level-crossing of the singlet-triplet excitation spectra11,12,13 , by taking account of the logarithmic corrections originated from the backward scattering. In this paper, we will analyze the following models based on this notion. The first example is the extended Hubbard model which is given by X ni ni+1 . (3) HEHM = HHM + V
I. INTRODUCTION
Spin-gap phases of one-dimensional (1D) electron systems have been studied for long times. This research has been motivated by the phase transitions in 1D organic conductors. In the past decade, the discovery of high-Tc superconductivity strongly stimulated this study. The spin-gap transition in 1D lattice models had been mainly analyzed by two approaches: The one is the weak coupling theory based on the bosonization theory and renormalization group. The other is numerical calculation in finite-size systems which is free from approximation. In the former scheme, the existence of the gap is argued by investigating the backward scattering effect on the fixed point, but the validity of the result is ensured only in the weak coupling limit. On the other hand, in numerical calculation, the analysis is done by a direct evaluation of the gap and the finite-size scaling method. In this approach, a singular behavior of the gap near the critical point makes it difficult to make out the instability. In order to illustrate the difficulty in the determination of the phase boundary, let us consider the Hubbard model, X X † ni↑ ni↓ . (1) (cis ci+1,s + H.c) + U HHM = −t is
i
i
For the study of spin-gap transitions, this model has been analyzed by the g-ology for weak coupling region7,8 . The numerical calculation was performed by the exact diagonalization with finite-size scaling method14,15,16 . However, the spin-gap phase boundary has not been clarified. The next example is the t-J model described by
This model has a spin gap for U < 0. According to the Bethe-ansatz result for the charge gap at half-filling1 combining a canonical transformation2 , we can obtain the asymptotic behavior of the spin gap near the critical point U = 0 as p ∆E ∼ 2t|U |e−πt/|U| . (2)
1
Ht-J = −t
X † (˜ cis c˜i+1,s + H.c.)
+J
quantum numbers. The second is derivation for the logarithmic corrections. The third explains the calculation in two-electron systems.
is
X i
(S i · S i+1 − ni ni+1 /4),
(4)
II. CONTINUUM FIELD THEORY
where c˜is = cis (1−ni,−s ). This model is obtained by doping holes in the Heisenberg spin chain. For this model, the weak coupling treatment is difficult due to this strong coupling constraint, however, the universality class of this model is known as TL liquids, from the analysis for the exactly solvable cases at J/t = 0 (spinless fermion) and J/t = 2 (super-symmetric point)17,18 . The remaining region was analyzed using the exact diagonalization by Ogata et al.19 . Their phase diagram shows the enhancement of the superconducting correlation (Kρ > 1) and the phase separation (Kρ → ∞) for the large J/t region. According to their result, the spin-gap phase does not exist except for the low density region. Variational approaches also played roles in the analysis for this model20,21,22 , but could not establish clear solution for the spin-gap phase. Extensions of the t-J model are also considered by many researchers23,24,25,26,27,28 . In spin systems, a spin gap opens by the effect of frustration or dimerization. Metallic spin-gap phases can be generated by doping holes in these spin systems. In this paper, we concentrate our attention on the t-J-J ′ model which includes the effect of frustration23,24 : X (S i · S i+2 − ni ni+2 /4). (5) Ht-J -J ′ = Ht-J + J ′
A. Effective Hamiltonian
Let us start our argument from the Abelian bosonization theory of electrons6,7,8,9,32 . The low-energy excitations are described by continuous fermion fields which are defined by cj,s → ψL,s (x) + ψR,s (x)
(6)
The boson representation of the fermion operator is ψr,s (x) = √
√ 1 eirkF x ei/ 2·[r(φρ +sφσ )−θρ −sθσ ] , 2πα
(7)
where α is a short-distance cutoff. r = R, L and s =↑, ↓ refer to +, − in that order. The phase fields are defined as √ iπ X 2πx φν (x) = − n ˆ ν , (8a) Ap (x) [νR (p) + νL (p)] − L L p6=0 √ 2πx iπ X m ˆ ν , (8b) Ap (x) [νR (p) − νL (p)] + θν (x) = + L L p6=0
where Ap (x) ≡ p1 e−iα|p|/2−ipx , and νr is the charge (ν = ρ) or the spin (ν = σ) density operator. These phase fields satisfy the relation [φν (x), θν (x′ )] = −iπ sign(x − x′ )/2. Using above relations, effective Hamiltonian of a 1D electron system is described by the U(1) Gaussian model (charge part) and the SU(2) sine-Gordon model (spin part), Z L √ 2g1⊥ (9) H = Hρ + Hσ + dx cos( 8φσ ). 2 (2πα) 0
i
We introduce a parameter α for the strength of the frustration given by α ≡ J ′ /J. At half-filling (n = 1), this model becomes an S = 1/2 frustrated spin chain. In this case, the ground state at α = 1/2 is the two-fold degenerate dimer state with a spin gap, and the ground state energy density is −3/4J 29,30,31 . The fluid-dimer transition occurs at αc = 0.241113. Upon doping of holes, the system may become metallic, and the spin gap is reduced23 but persists for the finite doping. The phase diagram of this model for n 6= 1 at α = 1/2, using the exact diagonalization, was obtained by Ogata, Luchini, and Rice24 , but the phase boundary of the spin-gap phase was also remained to be ambiguous. This paper is organized as follows. In Sec.II, we discuss, based on the continuum field theory, that the level crossing of singlet and triplet excitation spectra gives the critical point of the spin-gap transition. In Sec.III, we consider boundary conditions for the unique ground state, and discrete symmetries of wave functions to identify the energy spectra observed in our analysis. In Sec.IV, we analyze representative models introduced above, and clarify the spin-gap region in the phase diagrams, and check the consistency of our argument. Finally, in Sec.V, we present our conclusions. The paper also contains three Appendices. The first shows the relation among the different notations for the
Here g1⊥ is the backward scattering amplitude and for ν = ρ, σ Z vν L Hν = dx Kν (∂x θν )2 + Kν−1 (∂x φν )2 , (10) 2π 0 where vν and Kν are the velocity and the Gaussian coupling, respectively, for the charge (ν = ρ) and the spin (ν = σ) sectors. In the TL phase (g1⊥ > 0), the parameters Kσ and g1⊥ are renormalized as Kσ∗ = 1 and ∗ g1⊥ = 0, reflecting the SU(2) symmetry. The phase fields defined in eqs.(8) satisfy the following boundary conditions, √ (11a) φν (x + L) = φν (x) − 2πnν , √ θν (x + L) = θν (x) + 2πmν . (11b)
2
The quantum numbers mν and nν are defined by the ˆr,s (meaeigen values of the total number operators N sured with respect to the ground state) for right and left going particles (r = R, L) of spin s nν = [(NR↑ + NL↑ ) ± (NR↓ + NL↓ )]/2, mν = [(NR↑ − NL↑ ) ± (NR↓ − NL↓ )]/2.
or their linear combinations. From eqs.(7) and (11), the Fermi operator takes the following boundary conditions depending on the excited states: ψr,s (x + L) = ψr,s (x)eiπ(mρ +mσ +nρ +nσ ) .
(12a) (12b)
This means that the excited states given by arbitrary combination of quantum numbers are realized by changing the boundary conditions, while, for fixed boundary conditions, the possible excited states are restricted by the selection rule (13). The excitation spectra on which we will turn our attention can be obtained based on the operators for the charge-density-wave (CDW) and the spin-density-wave (SDW):
Here the upper and lower sign refer to charge (ν = ρ) and spin (ν = σ) degrees of freedoms, respectively. Thus nν denotes excitations involving the variation of particles numbers and mν indicate current excitations. If we require Nr,s to be an integer, the possible value of the quantum numbers are restricted as (−1)mρ ±mσ = (−1)nρ ±nσ .
(13)
This is the selection rule for the quantum numbers
6,35
† † OCDW = ψL↑ ψR↑ + ψL↓ ψR↓ √ √ 1 (21a) exp(i2kF x + i 2φρ ) cos( 2φσ ), = πα † † OSDW,z = ψL↑ ψR↑ − ψL↓ ψR↓ √ √ i exp(i2kF x + i 2φρ ) sin( 2φσ ), = (21b) πα † OSDW,+ = ψL↑ ψR↓ √ √ 1 exp(i2kF x + i 2φρ ) exp(+i 2θσ ). (21c) = 2πα
.
B. Excitation Spectra and Boundary Conditions
First, we consider the excitation spectra for g1⊥ = 0 case. If the system is periodic, and has unique ground state, the ground state energy of the system with length L is given by33 E0 (L) = Lǫ0 −
π(vρ + vσ ) c, 6L
(14)
These excitation spectra consist of the charge part which carries the momentum F , and the spin √ 2k√ part which forms singlet ( 2 cos 2φσ ) and triplet √ √ √ ( 2 sin 2φσ , exp(±i 2θσ )) states. Note that the spin part of the singlet and the triplet superconducting operators (SS, TS) are obtained with kF = 0 and replacing φρ → θρ . If charge-spin separation occurs, the spin excitations in eqs.(21) (mσ = 1 or nσ = 1, otherwise = 0) can be extracted by using anti-periodic boundary conditions following eq.(20). In the continuum field theory based on the TL model, the dispersion relation is approximated by linearized one, so that the deviation from the approximated dispersion become smaller if the excitation energies become lower by eliminating the charge excitations. Therefore, the precision of the analysis is enhanced by twisting the boundary conditions. The twisted boundary conditions are also important in identification of excitation spectra. Under anti-periodic boundary conditions, the momenta of these states are reduced to zero. Then we can define the parity transformation to classify these spectra. Although the spaceinversion operator and translation operator do not commute, we can classify these spectra simultaneously by wave numbers and parities, if the wave number k takes 0 or π. From eq.(7), the phase fields φν change under the parity (P: R↔L), and the spin-reversal transformations (T : ↑↔↓) as36
where the central charge c characterizes the universality class of the model. The finite-size corrections for the excitation energy and momentum of the system are described by34,6 2πvρ 2πvσ xρ + xσ , L L 2π P − P0 = (sρ + sσ ) + 2mρ kF , L
E − E0 =
(15) (16)
where kF = πN/2L is the Fermi wave number. xν = − + − ∆+ ν + ∆ν , sν = ∆ν − ∆ν are the scaling dimension and the conformal spin, respectively, where the conformal weights for each sector are given by !2 r K 1 n ν ν + n± (17) ∆± mν ± √ ν. ν = 2 2 2Kν Here the integer n± ν denote descendant fields which describe particle-hole excitations near the Fermi points. The scaling dimensions are related to the critical exponents for the correlation functions as hOi (r)Oi (r′ )i ∼ |r − r′ |−2(xρi +xσi ) .
(18)
Therefore, there is one to one correspondence between the excitation spectra and the operators. The operators correspond to the excited states are given by Omρ ,mσ ,nρ ,nσ ∝ ei
√
2(mρ φρ +mσ φσ +nρ θρ +nσ θσ )
,
(20)
P : φσ → −φσ , T : φσ → −φσ
(19) 3
φρ → −φρ
(22a) (22b)
Thus operators √ have discrete symmetries as P = T = 1 √ for the singlet ( 2 cos√ 2φσ ),√and P = T = −1 for the triplet with S z = 0 ( 2 sin 2φσ ). The discrete symmetries of the wave functions of these excited states are determined by combinations of those of the ground state and the operators. Further discussion for the discrete symmetries will be given in the next section.
Finally, let us consider the massive region. The behavior of the gap is explained from the two-loop renormalization group equation of the level-1 SU(2) WZNW model43,44,45 1 dy0 (l) = −y02 (l) − y03 (l). dl 2
If we define the correlation length ξ as y0 (ln ξ) ≡ −1 and the energy gap as ∆E = vσ /ξ, then one can derive the asymptotic form of the spin gap by solving the differential equation for |y0 (l)| ≪ 1 as p (26) ∆E ∝ |y0 | exp(−Const./|y0 |).
C. Renormalization Group
Next, we consider the renormalization (g1⊥ 6= 0). By the change of the cut off α → edl α, the coupling constant g1⊥ and Kσ are renormalized as37 dy0 (l) = −y12 (l), dl dy1 (l) = −y0 (l)y1 (l), dl
Note that eq.(26) is the same asymptotic behavior as the spin gap of the negative-U Hubbard model at half-filling given by eq.(2).
(23a) (23b)
III. UNIQUENESS OF GROUND STATE AND DISCRETE SYMMETRIES
where y0 (l) ≡ 2(Kσ − 1), y1 (l) ≡ g1⊥ /πvσ . For the SU(2) symmetric case y0 (l) = y1 (l) (the level-1 SU(2) Wess-Zumino-Novikov-Witten (WZNW) model38,39,11 ) and y0 (l) > 0, the scaling dimensions of the operators for singlet and triplet excitations split logarithmically by the marginally irrelevant coupling as40,11 (see Appendix B) 1 3 y0 + , 2 4 y0 ln L + 1 1 1 y0 = − , 2 4 y0 ln L + 1
(25)
xsinglet = σ
(24a)
xtriplet σ
(24b)
In the previous section, the ground state is assumed to be a singlet, so that we should consider the way to make the singlet ground state in the finite-size systems. Furthermore, we also discuss the discrete symmetries of wave functions to identify the energy spectra46 . The symmetries depend on the choice of representations for wave functions, so that we consider in the representative two cases: one is the standard electron systems such as the (extended) Hubbard model. The other is doped spin chains like the t-J model. In the following argument, the electron hopping is restricted to the nearest neighbor, and the number of electrons is assumed to be even. The results are summarized in Table I.
where y0 ≡ y0 (0) and we have set l = ln L. When y0 < 0, y0 (l) is renormalized to y0 (l) → −∞, then a spin gap appears. At the critical point (y0 = 0), there are no logarithmic corrections in the excitation gaps (the logarithmic correction from higher order also vanish). Therefore, the critical point is obtained from the intersection of the singlet and the triplet excitation spectra11,12,13 . In this case, we can determine the critical point with high precision13 , since the remaining correction is only xν = 4 irrelevant fields41,42 . This irrelevant field, which does not exist in the pure sine-Gordon model, comes from the nonlinear term neglected when linearizing the dispersion relation near the Fermi level in the course of the bosonization. The physical meaning of this transition point (y0 = 0) is the one where the backward scattering coupling changes from repulsive to attractive. Moreover, at the critical point, the SU(2) symmetry is enhanced to the chiral SU(2)×SU(2) symmetry11 , since the spin degrees of freedom of the right and the left Fermi points become independent. Eq.(24) also explains the fact that the SDW (CDW) correlation is dominant for Kρ < 1 region with(out) spin gap, while for Kρ > 1, the TS (SS) correlation is dominant with(out) spin gap40 .
A. Hubbard-type Models
It is convenient to use the following representation of the basis to describe the Hubbard-type models which permits double occupancy: X fA (n1 , · · · , nM ; nM+1 , · · · , nN ) |ΨA i ≡ n1 0 region15 . This is because the strong nearest-neighbor repulsion stabilizes the on-cite singlet pairs. The one of the striking feature in this phase diagram is that there are two phase separated states in the V /t ≫ 1 region, and the spin-gap phase boundary flows between these two phase-separated states. In this region, the spin-gap phase boundary shifts to the large U side due to the strong finite-size effect. The phase-separated state in the U/t > 0 side is considered as a mixture of 4kF - and 2kF CDW phases. The stability of this phase is already argued in Ref.15 by using the second-order perturbation theory. On the other hand, in the U/t < 0 side, the system is separated into a 2kF -CDW phase and a vacuum. These phase-separated states are illustrated in Fig.8. The consistency of the argument can also be checked as in the case of the t-J model. Fig.9 shows the averaged scaling dimension (41) at n = 1/2 for V /t = 2 and 8
APPENDIX A: QUANTUM NUMBERS IN TWO NOTATIONS
In the analysis of 1D electron systems by Bethe-ansatz results with CFT, a different notation from ours is often used to describe the quantum numbers17,18,35,50,51 . In these notation the spin degrees of freedom is imposed only on down spins. In their definition, ∆Nc is the change of the total number of electrons, and ∆Ns is the change of the number of down spins. Dc (Ds ) denotes the number of particles moved from the left charge (spin) Fermi point to the right one. They are given by the eigen value of the ˆr,s as number operator N ∆Nc ∆Ns 2Dc 2Dc + 2Ds
8
= NR↑ + NL↑ + NR↓ + NL↓ , = NR↓ + NL↓ , = NR↑ − NL↑ , = NR↓ − NL↓ .
(A1a) (A1b) (A1c) (A1d)
where the perturbation term LI consists of the following two parts: one is a part of Gaussian model which denotes the deviation from the free case (K = 1). The other is the cosine term which stems from the backward scattering. They denote the effect of interaction between the left and the right Fermi points, and are written in the Euclidean space as (B6a) O0 ≡ −α2 K −1 (v −1 ∂τ φ)2 + (∂x φ)2 , √ √ (B6b) O1 ≡ 2 cos 8φ.
From eqs.(12) and (A1), the quantum numbers can be read as nρ = ∆Nc /2, nσ = ∆Nc /2 − ∆Ns , mρ = 2Dc + Ds , mσ = −Ds . One can also easily show the equivalence of the selection rule given by eq.(13) and the one written by this notation35 : ∆Nc + ∆Ns (mod 1), 2 ∆Nc (mod 1). Ds = 2
Dc =
(A2a) (A2b)
This relation is derived from the U → ∞ limit of the Hubbard model.
Their coupling constants are given by √ 2λ0 ≡ y0 (l), 2λ1 ≡ y1 (l).
APPENDIX B: DERIVATION OF LOGARITHMIC CORRECTIONS
For the SU(2) symmetric case y0 (l) = y1 (l), and y0 (l) > 0, the marginally irrelevant coupling is calculated from eq.(23) as
Here we derive the logarithmic corrections given in eq.(24). Hereafter, we omit the spin index σ. We consider perturbation terms which break the scale invariance as X Z L dr H = H∗ − λi Oi (r). (B1) 0 2π i
y0 (l) =
x +xj −xk
r12i
x +xk −xi
r23j
x +xi −xj
r31k
(B8)
then the coefficients of their OPE with the marginal operators (B6) are obtained as
where |φi i is the eigen state of Ei , and Cijk is a universal constant (OPE (operator product expansion) coefficient) fixed by a three-point function: hOi (r1 )Oj (r2 )Ok (r3 )i =
y0 , y0 ln L + 1
where the the bare coupling is defined as y0 ≡ y0 (0), and we have set l = ln L. Now we consider the operators for the singlet and the triplet states as √ √ (B9a) O2 ≡ 2 cos 2φ, √ √ O3 ≡ 2 sin 2φ, (B9b) √ (B9c) O4 ≡ exp(+i 2θ),
Then the correction to the finite-size scaling is calculated within the first-order perturbation as41 X Z L dr 2πv Ei − E0 = xi − λj hφi |Oj (r)|φi i L 0 2π j xj −2 X 2π 2πv , λj Ciij xi − (B2) = L L j
Cijk
(B7)
K 1 , C440 = , 2 2K 1 = √ , C441 = 0. 2
C220 = C330 = − C221 = −C331 .
(B10)
Thus the scaling dimensions of the operators for singlet and triplet excitations are obtained from eqs.(B2), (B8), and (B10). These are consistent with the results obtained by Gimarchi and Schulz40 .
(B3) This coefficient can be derived from the following two ways.
2. Non-Abelian Bosonization 1. Abelian Bosonization
In the standard bosonization theory, systems are described in U(1) symmetric form, so that the explicit SU(2) symmetry in spin degrees of freedom is lost. In order to describe systems with higher symmetry, it is desirable to perform the calculation defining current fields that conserve the SU(2) symmetry. In SU(2) symmetric case, the system is described by chiral SU(2) currents that are defined as
The Lagrangian density of the spin part of eq.(9) (the sine-Gordon model) is written as L = L0 + LI
(B4)
with 1 −1 (v ∂τ φ)2 + (∂x φ)2 , 2π λ1 λ0 O0 + O1 , LI = 2πα2 2πα2 L0 =
(B5a)
† J r ≡: ψr,α
(B5b) 9
σ αβ ψr,β :, 2
(B11)
where σ = [σ 1 , σ 2 , σ 3 ] are the Pauli matrices and r = (R, L). The chiral SU(2) current J R has a conformal dimension (∆+ , ∆− ) = (1, 0) and J L has (∆+ , ∆− ) = (0, 1). The three components of J r obey commutation relations known as the Kac-Moody algebra with central charge k: Jri (z)Jrj (w) =
The result of the original Hamiltonian can be obtained when we set U = ∞ in the end of the calculation. It is well known for a two-body problem that the ground state is a singlet as far as the bottom of the energy band has no degeneracy56,57 . This is consistent with the argument in Sec.III. The wave function in this system can be written using the basis A as X f (i, j)c†i↑ c†j↓ |vaci (C2) |Ψi =
k/2 iεijl ∂Jrl (w) δij + + reg., (B12) 2 (z − w) z−w
ij
where εijl is the anti-symmetric structure factor. For spin-s systems, there is a relation k = s/2. If a system is described by this current algebra, the system belongs to the universality class of the Wess-ZuminoNovikov-Witten non-linear σ model with topological coupling k 38,39 . In this case, the scaling dimension is x=
where f (i, j) = f (j, i) for the singlet (T = 1) as shown in Sec.III. The Schr¨odinger equation for the singlet wave function is X Ef (i, j) = [til f (l, j) + tjl f (i, l)] + [U δij − Jij ]f (i, j), l
(C3)
2sr (sr + 1) , 2+k
(B13)
with tij = −tδ|i−j|,1 and Jij = Jδ|i−j|,1 +αJδ|i−j|,2 where α denotes the strength of the frustration α ≡ J ′ /J. The Fourier transformation of eq.(C3) is given by
where sr = 0, 1/2, · · · , k/2. Therefore, the lowest energy spectra for the singlet and the triplet excitations are xsinglet = xtriplet =
1 . 2
Ef (k1 , k2 ) = [t(k1 ) + t(k2 )]f (k1 , k2 ) 1X + [U − J(k)]f (k1 + k, k2 − k), L
(B14)
Now let us consider the correction in the presence of a marginal operator (x = 2)11 which is given by O = J L · J R,
(C4)
k
where
(B15)
f (k1 , k2 ) =
The marginal operator O is proportional to S L ·S R where S r is the SU(2) charge, and S = S L + S R is the spin of the state φi . Letting the degrees of S and S r are s and sr , respectively, the expectation value becomes
1X f (i, j)e−ik1 ri −ik2 rj , L ij
t(k) = −2t cos k, J(k) = 2J(cos k + α cos 2k).
(C5) (C6) (C7)
Next, we introduce center of mass and relative momenta by Q = k1 + k2 , q = (k1 − k2 )/2, and redefine the functions as
1 hφi |S L · S R |φi i = (s(s + 1) − sL (sL + 1) − sR (sR + 1)). 2 (B16)
fQ (k) ≡ f (k1 , k2 ), ǫQ (q) ≡ t(Q/2 + q) + t(Q/2 − q).
Here, sL = sR = 1/2 and s = 0 for the singlet and s = 1 for the triplet. Thus the ratio of the logarithmic corrections is calculated as 3 : −1.
(C8) (C9)
Then we get fQ (q) =
APPENDIX C: DILUTE LIMIT
U L
P
k
P fQ (k) − L1 k J(q − k)fQ (k) , E − ǫQ (q)
(C10)
where
In the low-density limit, a many-body problem may be reduced to a two-body problem. Here we consider a critical point where a bound electron pair become stable in the ground state, and a singlet-triplet level-crossing takes place. We perform the calculation following the approach of H. Q. Lin, which was used for the 2D case53 . In order to take the constraint of the t-J(-J ′ ) model into account, we relax the restriction, and add the on-site Coulomb term as X ˜ =H+U ni↑ ni↓ . (C1) H
J(q − k) = 2J(cos q cos k + α cos 2q cos 2k).
(C11)
Note that the terms that contain sin are omitted in eq.(C11), because they give no contribution due to their symmetry. Now we define the following variables, and iterate them as UX fQ (q) C0 ≡ L q = U I0,0 C0 − 2JU I1,0 C1 − 2αJU I0,1 C2 ,
i
10
(C12a)
C1 ≡
UX fQ (q) cos q L q
= I1,0 C0 − 2JI2,0 C1 − 2αJI1,1 C2 , UX C2 ≡ fQ (q) cos 2q L q = I0,1 C0 − 2JI1,1 C1 − 2αJI0,2 C2 ,
to the symmetry of the wave function: f (i, j) = −f (j, i) (T = −1). Therefore, the triplet state is always noninteracting. This means that the level-crossing point can be obtained as a solution (C16) for E = ǫQ=0 (π/L). In this case, the density dependence of the critical point of the t-J-J ′ model can be expanded as
(C12b)
(C12c)
Jc (n) = Jc (0) + A(α)n2 + O(n4 ),
where Im,n
1 X cosm q cosn 2q ≡ . L q E − ǫQ (q)
(C18)
where Jc (0) is same as the solution for the ground state. Therefore, the spin-gap phase boundary in the low-density limit coincides with the critical point for the bound electron pair in the ground state, and its curve is the square-root type in the J/t-n plane. For α = 0 case, we obtain eq.(40). These solutions reflect the shape of the band structure.
(C13)
The criterion that eq.(C12) have a solution is 1 − U I0,0 2JU I1,0 2αJU I0,1 2JI2,0 + 1 2αJI1,1 = 0, (C14) det −I1,0 −I0,1 2JI1,1 2αJI0,2 + 1 where all Im,n can be related to I0,0 as follows, I1,0 I2,0 I3,0 I4,0 I1,1 I0,1 I0,2
= (1 − EI0,0 )/4t, = −EI1,0 /4t, = (1 − 2EI2,0 )/8t, = −EI3,0 /4t, = 2I3,0 − I1,0 , = 2I2,0 − I0,0 , = I0,0 − 4I2,0 + 4I4,0 ,
∗
E-mail address:
[email protected] E-mail address:
[email protected] ‡ E-mail address:
[email protected] 1 A. A. Ovchinnikov, Zh. Eksp. Teor. Fiz. 57, 2137 (1969) [Sov. Phys. JETP 30, 1160 (1970)]. 2 H. Shiba, Prog. Theor. Phys. 48, 2171 (1972). 3 M. Nakamura, K. Nomura, and A. Kitazawa, Phys. Rev. Lett. 79, 3214 (1997). 4 M. Nakamura, J. Phys. Soc. Jpn. 67, 717 (1998). 5 S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950); J. M. Luttinger, J. Math. Phys. 4, 1154 (1963); D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 (1965). 6 F. D. M. Haldane, J. Phys. C 14, 2585 (1981). 7 V. J. Emery, in Highly Conducting One-Dimensional Solids, edited by J. T. Devreese et al. (Plenum, New York, 1979), p.327. 8 J. S´ olyom, Adv. Phys. 28, 201 (1979). 9 J. Voit, Rep. Prog. Phys. 57, 977 (1995). 10 N. Manyh´ ard and J. S´ olyom, J. Low Temp. Phys. 12, 529 (1973); J. S´ olyom, ibid 547 (1973). 11 I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys. A 22, 511 (1989). 12 T. Ziman and H. J. Schulz, Phys. Rev. Lett. 59, 140 (1987). 13 R. Julien and F. D. M. Haldane, Bull. Am. Phys. Soc. 28, 344 (1983); K. Okamoto and K. Nomura, Phys. Lett. A 169, 433 (1992); S. Eggert, Phys. Rev. B 54, 9612 (1996). 14 K. Penc and F. Mila, Phys. Rev. B 49, 9670 (1994). 15 ¯ K. Sano and Y. Ono, J. Phys. Soc. Jpn. 63, 1250 (1994). 16 H. Q. Lin, et al., The Hubbard Model, Edited by D. Baeriswyl et al., (Plenum Press, New York, 1995) p.315. 17 P. -A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990); P. -A. Bares, G. Blatter, and M. Ogata, Phys. Rev. B 44, 130 (1991). 18 N. Kawakami and S. K. Yang, Phys. Rev. Lett. 65, 2309 (1990); J. Phys. Condens. Matter 3, 5983 (1991). 19 M. Ogata, M. U. Luchini, S. Sorella, and F. F. Asaad, Phys. Rev. Lett. 66, 2388 (1991). 20 C. S. Hellberg and E. J. Mele, Phys. Rev. B 48, 646 (1993). †
(C15)
and I0,0 diverges. Then setting U = ∞, we get the relation between the singlet-state energy and the parameters of the model as 4t/J = −z(4αz 2 − 2α + 1) p + z 2 (4αz 2 − 2α + 1)2 − 4α(2z 2 − 1),
(C16)
where z ≡ E/4t. For the singlet pair with Q = 0, the energy is given by E = −4t + B where B is the binding energy. At the critical point where the singlet pair becomes stable, the binding energy becomes B = 0. Then we get the solution (44) without size dependence. In α = 0 case, we obtain Jc = 2t. In the case of the extended Hubbard model, the solution can be obtained by setting (J, α) = (−V, 0) in eq.(C14), and leaving U finite. The result is V =
2U . z(U/t − 4z)
(C17)
For U → 0 limit, it coincides with the spin-gap phase boundary and the Kρ = 1 contour line predicted by the g-ology: V = −U/2. Finally, we consider the singlet-triplet level-crossing point in the dilute limit. In the system with anti-periodic boundary conditions, the bottom of the energy band is degenerate, so that a level crossing may take place. For the triplet state, the last term of eq.(C4) vanishes due 11
21
(1991). M. Nakamura, in preparation. 56 J. C. Slater, H. Statz and G. F. Koster, Phys. Rev. 91, 1323 (1953). 57 For example, see K. Yosida, Theory of Magnetism, (Springer-Verlag, Berlin, Heidelberg 1996). -5.5
Y. C. Chen and T. K. Lee, Phys. Rev. B 47, 11548 (1993). H. Yokoyama and M. Ogata, Phys. Rev. Lett. 67, 3610 (1991); Phys. Rev. B 53, 5758 (1996). 23 K. Sano and K. Takano, J. Phys. Soc. Jpn. 62, 3809 (1993); K. Takano and K. Sano, Phys. Rev. B 48, 9831 (1993). 24 M. Ogata, M. U. Luchini and T. M. Rice, Phys. Rev. B 44, 12083 (1991). 25 M. Imada, J. Phys. Soc. Jpn. 60, 1877 (1991); Phys. Rev. B 48, 550 (1993). 26 E. Dagotto and J. Riera, Phys. Rev. B 46, 12084 (1992). 27 M. Troyer, H. Tsunetsugu, T. M. Rice, J. Riera, and E. Dagotto, Phys. Rev. B 48, 4002 (1993). 28 B. Ammon, M. Troyer, and H. Tsunetsugu, Phys. Rev. B 52, 629 (1995). 29 C. K. Majumder, J. Phys. C 3, 911 (1970); C. K. Majumder and D. K. Ghosh, J. Math. Phys. 10, 1399 (1969). 30 B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 47, 964 (1981). 31 P. M. van den Broek, Phys. Lett. A 77, 261 (1980). 32 H. J. Schulz, Phys. Rev. Lett. 64, 2831 (1990); Int. J. Mod. Phys. B 5, 57 (1991). 33 H. W. J. Bl¨ ote, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986); I. Affleck, Phys. Rev. Lett. 56, 746 (1986). 34 J. L. Cardy, J. Phys. A 17, L385 (1984). 35 F. Woynarovich, J. Phys. A 22, 4243 (1989). 36 In these transformation, we fix the number of electrons and of spins (nν = Const.). Therefore, we do not consider the variation of θν fields. 37 J. M. Kosterlitz, J. Phys. C 7, 1046 (1974). 38 V. Knizhnik and A. B. Zamolodchikov, Nucl. Phys. B 247, 83 (1984). 39 D. Gepner and E. Witten, Nucl. Phys. B 278, 493 (1986). 40 T. Giamarchi and H. J. Schulz, Phys. Rev. B 39, 4620 (1989). 41 J. L. Cardy, Nucl. Phys. B 270, 186 (1986). 42 P. Reinicke, J. Phys. A 20, 5325 (1987). 43 D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein, J. Phys. A 13, 585 (1980). 44 C. Destri, Phys. Lett. B 210, 173 (1988); ibid 213, 565E (1988). 45 K. Nomura, Phys. Rev. B 48 16814, (1993). 46 For example, in the Lanczos algorism, energy spectra can be classified according to symmetries by choosing the initial vector. 47 E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16 407 (1961); E. Lieb and D. Mattis, J. Math. Phys. 3 749 (1962). 48 M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 (1989). 49 W. Kohn, Phys. Rev. 133, A171 (1964); B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 65, 243 (1990). 50 N. Kawakami and S. K. Yang, Phys. Lett. A 148, 359 (1990). 51 H. Frahm and V. E. Korepin, Phys. Rev. B 42, 10553 (1990). 52 A system with twisted boundary conditions is transformed into PL a system with flux by the unitary operator exp(−iΦ j=1 jnj /L). In this transformation, the wave number k in the former system shifts to k + ΦN/L. 53 H. Q. Lin, Phys. Rev. B 44, 4676 (1991). 54 Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338
55
22
-6.0
4kF -CDW 2kF -CDW vσ
E
SDW vρ
-6.5
-7.0
GS -7.5
0.0
Singlet
I= 0 1 2 (2 k F) 3 4 (4 k F) 5 6 7 0.2
Triplet
0.4
Φ/π
0.6
0.8
1.0
FIG. 1. Spectral flow of the 1D t-J model at J/t = 2 with length L = 8 and electron number N = 4. These energy spectra are the lowest two levels for each wave number k = 2πI/L. The marked spectra correspond to the physical states written in this figure.
0.5 0.4
∆E/t
0.3 0.2 singlet triplet
0.1 0.0 0
1
J/t
2
3
FIG. 2. Singlet and triplet excitation energies of the 1D t-J model for L = 16 system at n = 1/2. These excitation spectra can be identified by symmetries of their wave functions.
12
1.0
2.710
(a) 0.8
2.708
Jc/t
2.706
?
TL
0.6
n
2.704
0:6
0:8
1: 0
+
N
1
42 4 2
0.4
2 4 6 8 10 12 14 16
PS
4 4 24
2.702
3 22 322 SG 3 33 3 33 3 3 3
0.2
2.700 2.698 0
0.005
2
1/L
0.01
1.0
0.015
TL 0.6
n
0:6
0:8
4 + ? c r
0.4
triplet singlet
1:0
4+
4 +2
+
+?
4
?
c
1
42
42 4 4 24 232 232 SG
0.8
0.7
3 2
(b) 0.8
FIG. 3. Size dependence of Jc /t determined by the intersections of the excitation spectra for L = 8, 12, 16, 20 systems at n = 1/2. These points are fitted by the form A+B/L2 +C/L4 .
0.2
PS
33 33 33 3 3 3
1.0
xσ
4+
4 2+
4 +
r c
0.6
(c) 0.8
0.5 n
0
1
2
J/t
3
0:6
0: 8
0.4
4 4 24
FIG. 4. Extrapolated value of (xsinglet +3xtriplet )/4 and the σ σ scaling dimensions for the singlet and the triplet excitations for L = 16 system at n = 1/2.
?+
TL
0.6
0.4
SG
42
42
4+
4 +2
+
1:0
0.0 0
1
1 PS
2 223 323 3 33 33 33 3 3
0.2
2
3
4
5
J=t
0.510 0.508
FIG. 6. Phase diagrams of the 1D t-J-J ′ model at (a) α = 0, (b) α = αc , (c) α = 1/2 (TL: TL phase, SG: spin-gap phase, PS: phase-separated state). In the spin-gap phase where the backward scattering is attractive, the singlet excitation becomes lower than the triplet (see FIG.2,4). The contour lines of Kρ are calculated by the data of L = 16 system.
J/t=2.0 J/t=2.2 J/t=2.4
xσ
0.506 0.504 0.502 0.500 0
0.005
0.01
0.015
2
1/L
FIG. 5. Size dependence of the averaged scaling dimension (xsinglet + 3xtriplet )/4 at n = 1/2. σ σ
13
10 8
(a) n=1/3
0.7
6
singlet triplet 0.6
4 2
xσ
V/t
SDW CDW
SS
0.5
0 TS
PS
-2 10 8
PS
0.4
TS
(b) n=1/2 SS
CDW (ins)
4
0.8
(b)
0.7
singlet triplet
CDW
xσ
V/t
6
2
0.6
SDW SS
0
0.5
-2
TS
PS
0.4
10 8
PS
(c) n=2/3
-1.0
PS
6
SS
4
-0.5
0.0
U/t
0.5
1.0
1.5
FIG. 9. Extrapolated value of (xsinglet +3xtriplet )/4 and the σ σ scaling dimensions for the singlet and the triplet excitations for L = 16 system at n = 1/2. (a) is the case of V /t = 2, and (b) is the case of V /t = 8.
TS
V/t
(a)
CDW SDW
2 0
SS
-10
TS
PS
-2 -8
-6
-4
-2
0
U/t
2
4
6
8
Ground state Singlet Triplet (S z = 0) Triplet (S z = 1)
10
FIG. 7. Phase diagrams of the 1D extended Hubbard model determined by the data of L = 12 systems at (a) n = 1/3, (b) n = 1/2, (c) n = 2/3 (SDW (TS): TL liquid phase with Kρ < 1 (Kρ > 1), CDW (SS): spin-gap phase with Kρ < 1 (Kρ > 1), PS: phase-separated state).
basis A P T 1 1 1 1 −1 −1 1 ∗
basis P ±1 ±1 ∓1 ∓1
B T ±1 ±1 ∓1 ∗
k 0 0 0 0
BC ∓1 ±1 ±1 ±1
TABLE I. Discrete symmetries of wave functions for different two bases (P: space inversion, T : spin reversal, k:wave number BC: boundary conditions). The upper (lower) sign denotes the case of N/2 =even (odd). These correspondences are explained by the Perron-Frobenius theorem and the bosonization theory.
(a) (b) FIG. 8. Two phase-separated states which appear in the V /t ≫ 1 region of Fig.7(c). (a) is located in U/t > 0 side. (b) lies in U/t < 0 side. Spin-gap phase boundary exists between these two phase-separated states in the phase diagram.
14
